Bramble, 31, is out of contract in June and there have been no talks over a new deal.
Despite injuries, error-ridden displays and off-field problems, Bramble has won back his place under Martin O’Neill.
But the Black Cats boss wants to reshape his defence and Boro boss Tony Mowbray will be alerted to Bramble’s availability.
Mowbray played alongside and coached Bramble in his early days at Ipswich and the defender would be keen on a switch to Boro.